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Article Contents

# Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations

• Corresponding author: M. Monir Uddin (E-mail: monir.uddin@northsouth.edu)
• To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations. In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established for solving the Lyapunov equations of the standard or generalized state space systems. In this paper, we develop algorithms for solving the Lyapunov equations for large-sparse structured descriptor system of index-1. The resulting algorithm is applied for the balancing based model reduction of large sparse power system model. Numerical results are presented to show the efficiency and capability of the proposed algorithm.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  Convergence histories of both Gramians by RKSM for mod-2

Figure 2.  Comparison between full system and reduced-order system in frequency domain

Figure 3.  Comparison between original system and reduced model in time domain

Figure 4.  Largest Hankel singular values of original system and 86 dimensional reduced-order system

Table 1.  Number of differential & algebraic variables and largest eigenvalue of $(-A, E)$ for different models.

 Model differential algebraic eigs$(-A, E)$ inputs/outputs Mod-1 606 6 529 10727 4/4 Mod-2 1 142 8 593 10727 4/4 Mod-3 3 078 18 050 10669 4/4

Table 2.  Comparisons between full systems and their reduced models

 Model Dimension Error full reduced absolute relative Mod-1 $7\, 135$ $87$ $3.1\times 10^{-3}$ $1.5 \times 10^{-4}$ Mod-2 $9\, 735$ $86$ $5.3 \times 10^{-2}$ $4.7 \times 10^{-4}$ Mod-3 $21\, 128$ $77$ $5.6 \times 10^{-1}$ $4.3 \times 10^{-2}$

Table 3.  Balanced truncation tolerances and dimensions of reduced-order model.

 Model tolerance dimension of ROM $10^{-4}$ 118 $10^{-3}$ 104 Mod-2 $10^{-2}$ 86 $10^{-1}$ 70

Figures(4)

Tables(3)